kitchen sinks always have a flat bottom, which allows anything put in the sink to stand up.
bathroom sinks are usually distinct from that, by having a rounded bottom, which makes any speck of skin slide more readily into the drain.

OTher than that, anything goes. One can buy salad bowls, drill holes in them and install them. Cost $10. It's been done before; this is not unique, new or risky. Hope this helps.

A sink with an overflow hole needs a drain that has an overflow hole too.
A sink with no overflow hole needs a drain that has no overflow hole.
There are slight advantages and disadvantages to each.
Hope this helps.

Bee, I think a rectangular sink or the one in the last picture I've posted would look fabulous in your bathroom. The style would sort of continue the knack you seem to have (and I love) which is creating spaces that are a little different and special. Lowe's has this one for less than a 100.00

I love the way the rectangular shape looks with the marble:

Here is a sample of a round vessel in a bathroom designed by Sarah Richardson

There are a ton of options in terms of sink styles; undermount, self-rimming, vessel, and more. While I love many of the options, my personal preference is for undermount sinks. I find them to be clean and classic and both showcase the countertop (especially marble) and maximize countertop surface. While I do like some of the unique shapes, I find circular to be most pleasing and classic and again able maximize countertop space.

For your unique home, you might consider a more interesting shape as lukkiirish suggests. Here is a unique shape from Kohler.

Similarly there are multiple material options including cast iron, stainless steel, glass, china, marble, and I'm sure many more. Unfortunately I have not done much research on materials and various benefits and drawbacks. Hopefully someone else can provide some useful information.